140-year-old math problem solved by researcher

March 3, 2008

A problem which has defeated mathematicians for almost 140 years has been solved by a researcher at Imperial College London.

Professor Darren Crowdy, Chair in Applied Mathematics, has made the breakthrough in an area of mathematics known as conformal mapping, a key theoretical tool used by mathematicians, engineers and scientists to translate information from a complicated shape to a simpler circular shape so that it is easier to analyse.

This theoretical tool has a long history and has uses in a large number of fields including modelling airflow patterns over intricate wing shapes in aeronautics. It is also currently being used in neuroscience to visualise the complicated structure of the grey matter in the human brain.

A formula, now known as the Schwarz-Christoffel formula, was developed by two mathematicians in the mid-19th century to enable them to carry out this kind of mapping. However, for 140 years there has been a deficiency in this formula: it only worked for shapes that did not contain any holes or irregularities.

Now Professor Crowdy has made additions to the famous Schwarz-Christoffel formula which mean it can be used for these more complicated shapes. He explains the significance of his work, saying:

"This formula is an essential piece of mathematical kit which is used the world over. Now, with my additions to it, it can be used in far more complex scenarios than before. In industry, for example, this mapping tool was previously inadequate if a piece of metal or other material was not uniform all over - for instance, if it contained parts of a different material, or had holes."

Professor Crowdy's work has overcome these obstacles and he says he hopes it will open up many new opportunities for this kind of conformal mapping to be used in diverse applications.

"With my extensions to this formula, you can take account of these differences and map them onto a simple disk shape for analysis in the same way as you can with less complex shapes without any of the holes," he added.

Professor Crowdy's improvements to the Schwarz-Christoffel formula were published in the March-June 2007 issue of Mathematical Proceedings of the Cambridge Philosophical Society.

Related Stories

(PhysOrg.com) -- Across the universe, galaxies band together in clusters so huge it can take 10 million years for light to travel from one end of a galaxy cluster to the other. Probing these metropolises is no easy task. ...

In the aftermath of World Breastfeeding Week, leading academics in infant nutrition from the Australian National University, Julie Smith, Libby Salmon and Phillip Baker, examine the challenges that remain in keeping breastfeeding ...

A team of neuroscientists report in the July 21 issue of the science journal Neuron their research about how neuron clusters in the brain overlap to communicate such combined visual information as a flower's color, shape,and ...

Science from the space shuttle helped open Earth's eyes to the cosmos and sister planets. It created perhaps the most detailed topographical map of Earth. And it even is helping doctors understand, and sometimes fix, what's ...

Since 2008, MIT economist Tavneet Suri has studied the financial and social impacts of Kenyan mobile-money services, which allow users to store and exchange monetary values via mobile phone. Her work has shown that these ...

Researchers have discovered a dinosaur tail complete with its feathers trapped in a piece of amber. The finding reported in Current Biology on December 8 helps to fill in details of the dinosaurs' feather structure and evolution, ...

Reporting new research results involves detailed descriptions of methods and materials used in an experiment. But when a study uses computers to analyze data, create models or simulate things that can't be tested in a lab, ...

Nothing ruins a potentially fun event like putting it on your calendar. In a series of studies, researchers found that scheduling a leisure activity like seeing a movie or taking a coffee break led people to anticipate less ...

(Phys.org)—Douglas Petrovich, an archaeologist with Ontario's Wilfrid-Laurier University in Canada has sparked controversy in the ancient history scholarly community by making claims that he has found proof that Hebrew ...